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PHYSICS 244 NOTES
Lecture 9
Comparing with experiment
Introduction
In classical mechanics, with the motion of a particle described by
its trajectory function r(t), comparing theory and experiment is
pretty straightforward. We set our watch and our coordinate
system. At time t we look for the particle at the point r(t).
In quantum mechanics, it is not quite so simple. We have ψ(x,t),
but this is not a very direct description. We have to perform some
operations on ψ, and then do repeated experiments. The one
example of this that we have already discussed is a position
measurement. We first take |ψ(x,t)|2 , and then do the experimental
preparation, wait for a time t, measure the position, and repeat
many times. After many repetitions, we build up a histogram for
x, and compare it to |ψ(x,t)|2 . There are similar prescriptions for
other quantities besides the histogram of x, and these are the
subject of this lecture.
Expectation values
The reason for the repetition is that quantum mechanics does not
make definite predictions for the position, momentum, etc. When
we do the exact same measurement on identically prepared
systems, we do not get always get the same result, as we do in
classical mechanics.
But probability distributions can be built up by repeated
measurements, and ψ does tell us what these should be. That is the
histogram approach. However, very often we are not interested in
the whole distribution, but only some of its simpler characteristics,
such as the average.
Position
We have already considered one example of this. <x>, the average
value of the position, can be obtained by
<x> = ∫ |Ψ(x,t)|2 x dx
For a particle in the ground state of the infinite square well, we
have
<x> =
∫ (2/L)1/2 sin(nπx/L) exp(iωt) (2/L)1/2 sin(nπx/L) exp(-iωt) x dx
= (2/L) ∫0L sin2(nπx/L) x dx
= (2/L) (L/nπ)2 ∫0nπ sin2 u u du
= (2L/n2π2) (n2π2/4) = L/2
This makes sense – it says that the particle is, on average, at the
middle of the well.
Kinetic and potential energy
The Schrödinger equation for a stationary state is
Eψ(x) = - (ħ2/2m) d2 ψ(x) / dx2 + V ψ(x).
Multipliying this by ψ* and integrating, we have
E∫ |ψ(x)|2 dx = - (ħ2/2m)∫ ψ*(x)(d2 ψ(x) / dx2)dx + ∫ V(x) |ψ(x)|2 dx,
and since ∫ |ψ(x)|2 dx = 1, we find
E = - (ħ2/2m) ∫ ψ*(x) (d2 ψ(x) / dx2) dx + ∫ V(x) |ψ(x)|2
= <KE> + <PE>.
So the average value of the kinetic energy is
<KE> = - (ħ2/2m) ∫ ψ*(x) (d2 ψ(x) / dx2) dx and
the average value of the potential energy is
<PE> = ∫ V(x) |ψ(x)|2 dx.
Momentum
Using the formula for the average kinetic energy:
KE = - ( ħ2/2m) ∫ ψ* d2ψ/dx2 dx = p2 / 2m
and roughly taking the square root, we “derive” the formula
<p> = - i ħ ∫ ψ* (dψ/dx) dx
for the average momentum of a stationary state.
Expectation values in the infinite square well
We already did the position. Here is the averagemomentum in the
state ψn:
<p> = -i ħ ∫ (2/L)1/2 sin(nπx/L) exp(iωt)
× (2/L)1/2 (d/dx) sin(nπx/L) exp(-iωt) dx
= -i ħ (2/L) ∫ sin(nπx/L)exp(iωt) (d/dx)sin(nπx/L) exp(-iωt) dx
= -i ħ (2/L) ∫ sin(nπx/L) cos(nπx/L) dx
= -i ħ (2/L) (L/nπ) ∫0nπ sin u cos u du
= -i ħ (2/nπ) ∫0nπ sin 2u du /2
= -i ħ (2/nπ) (1/4) [cos 2nπ-cos 0]
=0.
This is consistent with our characterization of these states as
consisting of an equal mixture of + and – momenta. Just because
the average momentum vanishes does not mean that the average
squared momentum need vanish:
<p2> = - ħ2 ∫ (2/L)1/2 sin(nπx/L) exp(iωt)
× (2/L)1/2 (d2/dx2) sin(nπx/L) exp(-iωt) dx
= - ħ2 (2/L) ∫ sin(nπx/L) (d2/dx2) sin(nπx/L) dx
= ħ2 (2/L) (nπ/L)2 ∫0L sin2(nπx/L) dx
= ħ2 (2/L) (nπ/L)2 (L/2)
= (ħ n π / L)2.
This may appear to be paradoxical at first: how can the average
momentum be zero, but the average squared momentum be
positive? But remember, the momentum distribution shows an
equal probability of having positive and negative momentum, but
the squares of these are always positive!
The expectation value of the kinetic energy in the state ψn is
<KE> = <p2 > / 2m = (ħ2 n2 π2 / 2m L2)
The big difference between the calculations for x and p is that
when we calculate p we need to change the wavefunction by taking
the derivative. For this reason, pop = - i ħ d/dx is called the
momentum operator – it operates on the wavefunction ψ to
produce a different function –i ħ dψ/dx.
Actually, however, x also operates on the wavefunction to produce
a new function xψ.
x is a multiplication operator, while p is a differential operator.
H = -ħ2 / 2m d2/dx2 + V is a mixed operator. It is called the
Hamiltonian or the energy operator.